Download Section 2.5 - Warren County Public Schools

2.5 The Fundamental Theorem of
Algebra
The Theorem
• If the degree of a polynomial is n, then there is
at least 1 zero is in the complex number
system
– All real numbers are in the complex number
system
Linear Factorization
• If a polynomial has degree n, then there are
exactly n linear factors
f ( x)  a ( x  c )( x  c )...(x  c )
n
1
2
n
Examples
f ( x)  x  6 x  9
2
g ( x)  x  4 x
3
Examples
• Write the polynomial as a product of factors
and list all zeros.
h( x)  x  x  2 x  12 x  8
5
3
2
Conjugate Pairs
• If (a+bi) is a zero, then (a-bi) is a zero.
Example
• Find the 4th degree polynomial that has -1, -1,
and 3i as zeros.
Prime Factors
• Also called Irreducible over the Reals
x  1  ( x  i )( x  i )
2
x  2  ( x  2 )( x  2 )
2
Examples
Completely factor
f ( x)  x  x  20
4
2
Examples
• Find all zeros of
f ( x)  x  3x  6 x  2 x  60
4
If (1+3i) is a zero.
3
2
Homework
• Page 144-145
1-4 all, 11-27 mult. of 3, 37, 39, 47
Worksheet